A Formula for the M\"obius function of the Permutation Poset Based on a Topological Decomposition

TL;DR

This paper introduces a simplified two-term formula for calculating the M"obius function of permutation intervals, leveraging normal occurrences and conjecturing the second term often vanishes, thus enabling more efficient computations.

Contribution

It provides a novel, simplified formula for the M"obius function in permutation posets, reducing computational complexity and offering insights into when the second term is zero.

Findings

The first term can be computed in polynomial time.

The second term often vanishes, simplifying calculations.

The formula applies to various cases, including poset fibrations.

Abstract

We present a two term formula for the M\"obius function of intervals in the poset of all permutations, ordered by pattern containment. The first term in this formula is the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the M\"obius function of this and other posets, but simpler than most of them. The second term in the formula is complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the M\"obius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. We also present a result on the M\"obius function of posets connected by a poset fibration.